Page 248 - DMTH403_ABSTRACT_ALGEBRA

Unit 25: Roots of a Polynomial

This definition of the product is consistent with what we would expect to obtain using a naive Notes
approach: Expand the product using the distributive law repeatedly (this amounts to multiplying
each term be every other) and then collect similar terms.
Proposition: If f(x) and g(x) are non-zero polynomials in F[x], then f(x)g(x) is non-zero and
deg(f(x)g(x)) = deg(f(x)) + deg(g(x)).
Corollary: If f(x),g(x),h(x) are polynomials in F[x], and f(x) is not the zero polynomial, then
f(x)g(x) = f(x)h(x) implies g(x) = h(x).
Definition: Let f(x),g(x) be polynomials in F[x]. If f(x) = q(x)g(x) for some q(x) in F[x], then we say
that g(x) is a factor or divisor of f(x), and we write g(x) | f(x). The set of all polynomials divisible
by g(x) will be denoted by < g(x) >.
Lemma: For any element c in F, and any positive integer k,
(x - c) | (x - c ).
k
k
Theorem 3: Let f(x) be a non-zero polynomial in F[x], and let c be an element of F. Then there
exists a polynomial q(x) in F[x] such that
f(x) = q(x)(x - c) + f(c).

Moreover, if f(x) = q (x)(x - c) + k, where q (x) is in F[x] and k is in F, then q (x) = q(x) and k = f(c).
1
1
1
Definition: Let f(x) = a x + · · + a belong to F[x]. An element c in F is called a root of the
m
0
m
polynomial f(x) if f(c) = 0, that is, if c is a solution of the polynomial equation f(x) = 0 .
Corollary: Let f(x) be a non-zero polynomial in F[x], and let c be an element of F. Then c is a root
of f(x) if and only if x-c is a factor of f(x). That is,
f(c) = 0 if and only if (x-c) | f(x).
Corollary: A polynomial of degree n with coefficients in the field F has at most n distinct roots
in F.

Self Assessment

1. Let F be a field and f(x)  F[x] then we say that an element a  F is a root of f(x) of
f(n) = ...............
(a) 1 (b) 2
(c) 0 (d) 2 -1

2. 1 is a root of x – 1  R[x], since 1 – 1 = ...............
2
2
(a) 2 (b) 1
(c) 0 (d) –1

1 1
3. ............... is a root of f(x) = x + x + 2 x  2  Q[x]
3
2
(a) 1 (b) 2
(c) –1 (d) –2
4. If n = 0 then f(x) is a non-zero ............... polynomial
(a) constant (b) degree

(c) range (d) power

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